I recently built a simple 22watt amp with the tda1554q chip for my first project. I am now looking to build some simple speaker boxes and purchase speakers for it. I'm wondering if anyone has suggestions on decent speakers for this amp? I was thinking 5 1/4" 3 way car speakers? do they even sell "home audio" speakers separate from the box? I'm not looking to spend very much on these speakers maybe 50$ max because I'm planning on building a 150 watt chip amp after this one for my main project.

There is a Loudspeakers section in the Forum with many interesting projects. Especially speakers are a difficult thing to recommend. Personal tastes are different and speakers have a much bigger impact on the sound than amplifiers do. Speakers also interact strongly with the room they play in, therefore you should choose them accordingly. Last but not least you should factor in the type of music you listen to and find a speaker with a fitting character.

With your budget you should look into Cyburgs' projects like Viech or Needle to name only two.

The Madisound Recession Buster kit with Vifa drivers is slightly above your
budget, but the speakers sound pretty good with my Mauro Penasa MyRef amp.
I understand there is a ver 2 kit which they have now.